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An orthocomplemented lattice is complemented. (def) 8. A complemented lattice is bounded. (def) 9. An algebraic lattice is complete. (def) 10. A complete lattice is bounded. 11. A heyting algebra is bounded. (def) 12. A bounded lattice is a lattice. (def) 13. A heyting algebra is residuated. 14. A residuated lattice is a lattice. (def) 15. A ...
The map φ + is a lattice isomorphism from L onto the lattice of all compact open subsets of (X,τ +). In fact, each spectral space is homeomorphic to the prime spectrum of some bounded distributive lattice. [3] Similarly, if φ − (a) = {x∈ X : a ∉ x} and τ − denotes the topology generated by {φ − (a) : a∈ L}, then (X,τ −) is ...
In mathematics, a supermodular function is a function on a lattice that, informally, has the property of being characterized by "increasing differences." Seen from the point of set functions, this can also be viewed as a relationship of "increasing returns", where adding more elements to a subset increases its valuation.
In mathematics, Mahler's compactness theorem, proved by Kurt Mahler , is a foundational result on lattices in Euclidean space, characterising sets of lattices that are 'bounded' in a certain definite sense. Looked at another way, it explains the ways in which a lattice could degenerate (go off to infinity) in a sequence of lattices
In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice.
A join-semilattice is bounded if it has a least element, the join of the empty set. Dually , a meet-semilattice is bounded if it has a greatest element , the meet of the empty set. Other properties may be assumed; see the article on completeness in order theory for more discussion on this subject.
Another famous example is the BCS theory of superconductivity. Here the underlying theory is the theory of electrons in a metal interacting with lattice vibrations called phonons. The phonons cause attractive interactions between some electrons, causing them to form Cooper pairs. The length scale of these pairs is much larger than the ...
An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force resulting from their opposite charges. It is an electrically neutral quasiparticle regarded as an elementary excitation primarily in condensed matter, such as insulators, semiconductors, some metals, and in some liquids.